2012-05-29

Derived Ritz Vectors

MDOF Numerical Integration

Multiple Support Excitation

Today a sort of miscellanea to close the topic of MDOF systems.

  1. Derived Ritz Vectors

    The Derived Ritz Vectors follows from a combination of matrix iteration and Gram-Schmidt orthogonalisation.

    Remarkable properties of DRVs are:
    • orthogonalisation is needed only with respect to the two previous DRVs
    • one of the two sweeping coefficients needed was already computed as a normalization factor in the preceding step
    • the reduced, Ritz coordinates eigenproblem can be easily formulated in canonical form with respect to a tridiagonal matrix
    • the equation of motion for Ritz coordinates can be written with a tridiagonal mass matrix, a unit stiffness matrix and a load vector that has only one coefficient different from zero
    • even for complex load shape, the expansion of load shape in terms of DRVs requires only a few lower terms, removing the requirement for static correction that, for these loadings, is needed in modal superposition procedure
  2. Algorithms for the Numerical Integration of MDOF systems

    Main problem in numerical integration of the equation of motion for a MDOF system is the numerical stability of the algorithm used. In short, only unconditionally stable algorithms can be used.
    It turns out that -formally- the algorithms are identical to the algorithms discussed in the context of SDOF systems, with identical numerical coefficients and dimensional scalar factors substituted by corresponding matrices.
    • the constant acceleration algorithm is unconditionally stable, we discussed a pseudocode listing that implements the c-a algorithm
    • a variation on the linear acceleration algorithm, introduced by Wilson and named Wilson's theta method, is unconditionally stable
    As direct numerical integration is usually needed for non-linear system, finally we reviewed the modified Newton-Raphson procedure.
  3. Multiple Support Excitation

    Multiple support excitation occurs in different cases, e.g., multi-span bridges, vehicles, equipment contained in vibration affected environments, etc. We recognized that in the vast majority of cases the equation of motion can be written in terms of the supports' accelerations only.

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Giacomo Boffi